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Riemann curvature tensor

For a type-$(2,0)$ tensor, the <Ricci identity> acts once on each contravariant index:
$$
[\nabla_\alpha,\nabla_\beta]T^{\mu\nu}
=R^\mu{}_{\rho\alpha\beta}T^{\rho\nu}
+R^\nu{}_{\rho\alpha\beta}T^{\mu\rho}.
$$

Curvature commutator on a contravariant tensor

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 1I / b / Solution

Curvature commutator on a contravariant tensor

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Curvature commutator on a contravariant tensor

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